2. Ambiguous Rings Based on a Polygon

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This Demonstration explores ambiguous rings.
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Contributed by: Erik Mahieu (May 2018)
Open content licensed under CC BY-NC-SA
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Details
The parametric equation of a circular cylinder with radius inclined at an angle
from the vertical is given by:
,
with parameters and
.
Define the functions:
and
.
The and
functions define the composite curve of the
-gonal base of the polygonal cylinder [1].
The parametric equation of a polygonal cylinder with sides and radius
rotated by an angle
around its axis is given by:
,
with parameters and
.
To find the equation of the intersection curve, set the corresponding components equal. This gives the three equations:
,
,
,
with
.
These are equations with four variables, ,
,
and
. Eliminating
,
and
by solving the equations gives the parametric equation of the intersection curve with θ as the only parameter:
.
Reference
[1] E. Chicurel-Uziel, "Single Equation without Inequalities to Represent a Composite Curve," Computer Aided Geometric Design, 21(1), 2004 pp. 23–42. doi:10.1016/j.cagd.2003.07.011.
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